RDF Surfaces Primer

1. Introduction

This document presents a Notation3 sublanguage to express a revised RDF logic as envisoned by Pat Hayes in his 2009 ISWC Invited Talk: BLOGIC. RDF Surfaces logic can be thought as an implementation of Charles Sanders Peirce's Existential Graphs in RDF.

An RDF surface is a kind of a sheet of paper on which RDF graphs can be written. All triples that are part of an RDF graph are then on this sheet of paper, including all [URI]s, literals and Blank nodes. A sheet of paper can contain more than one RDF graph. An RDF graph can’t be split over multiple sheets of paper. But, one can copy an RDF graph from one sheet of paper to another sheet of paper.

Blank nodes are special. They can’t be copied to a new piece of paper. They can be thought of scratches on a piece of paper. These scratches, or graffiti, are engraved into the piece of paper and can’t be transferred. When copying data with blank nodes onto a new sheet of paper, new scratches need to be made.

@prefix : <http://example.org/ns#> .

[] a :City .
[] a :Cat .

Positive surface

There are special sheets of paper that can be interpreted as a logical statement. For instance, the sheet of paper in Example 1 is an example of a positive surface (we use in this document a paper with a black border as a positive surface). Any RDF graph written on this surface is interpreted as logical assertion (true). An empty sheet of positive paper is an empty claim and is treated as a logical tautology (true).

When there is there is more than one RDF graph on the surface, then it is a logical conjunction (AND). If we interpret the sheets of paper with the black border in the examples above as positive surfaces, then they express:

• Example 1: true

• Example 2: It is true that: Ghent a City.

• Example 3: It is true that: ∃ x : x a City

• Example 4: It is true that: ∃ x , y : ( x a City ) AND ( y a Cat )

The blank nodes need to be interpreted as existential quantified variables on logical paper.

A positive surface is the default surface, when no other surface information is provided.

Negative surface

In the same way as a positive surface asserts a logical truth, a negative surface asserts a logical negation. The examples below will use a sheet of paper with a red border as a negative surface.

An empty negative surface on the default positive surface expresses a logical contradicton. When there are one or more RDF graphs written on an negative surface they mean the negation of those RDF graphs.

A blank node on a negative surface is interpreted as a universal quantified variable. The reason is that:

NOT(∃ x : P(x)) ⇔ ∀ x : NOT(P(x))

Propositional logic

With combination of conjunction, by putting triples on a positive surface, and negation, by putting triples on a negative surface, any compound truth-functional statement can be symbolized with positive and negative sheets of paper:

• The logical conjunction AND is given by putting RDF triples on a positive surface.

• The logical negation NOT is given by putting RDF triples on a negative surface.

• The logical disjunction OR can be made by a combination of positive and negative surfaces:

  • NOT(P AND Q) = NOT(P) OR NOT(Q)

  • NOT( NOT(P) AND NOT(Q)) = P OR Q

• The logical implication → can be made with a combination of AND and OR:

  • P → Q = NOT(P) OR Q = NOT( P AND NOT(Q) )

First-order logic

First-order logic can be added to the RDF surfaces by interpreting the blank node as an existential quantified variable and using the rule that a universal quantified variable can made from an existential quantified variable by placing it in an enclosing negative surface:

∀ x : P(x) ⇔ NOT(∃ x : NOT P(x)):

• A blank node on a positive surface references an existential quantified variable.

• A blank node on a negative surface references a universal quantified variable.

First-order logic in Notation3

RDF Surfaces provides a way to express the notion of positive and negative surfaces in Notation3 with help of the built-ins log:onPositiveSurface and log:onNegativeSurface.

The subject of both built-ins are the blank nodes that need to be put on the surface (so that they become existential or universal quantified variables depending on the type of surface).

@prefix : <http://example.org/ns#> .
@prefix log: <http://www.w3.org/2000/10/swap/log#> .

(_:X) log:onPositiveSurface {
    _:X a :City .
} .

@prefix : <http://example.org/ns#> .
@prefix log: <http://www.w3.org/2000/10/swap/log#> .

_:X a :City .

@prefix : <http://example.org/ns#> .
@prefix log: <http://www.w3.org/2000/10/swap/log#> .

(_:X) log:onNegativeSurface {
    _:X a :Problem .
} .

@prefix : <http://example.org/ns#> .
@prefix log: <http://www.w3.org/2000/10/swap/log#> .

(_:X) log:onNegativeSurface {
    _:X a :Cat .

    () log:onNegativeSurface {
        _:X :is :Alive .
    } .

    () log:onNegativeSurface {
        _:X :is :Dead .
    } .
} .

@prefix : <http://example.org/ns#> .
@prefix log: <http://www.w3.org/2000/10/swap/log#> .

(_:X) log:onNegativeSurface {
    _:X a :Cat ;
        :is :Alive .
    () log:onNegativeSurface {
        _:X :says :Meow .
    } .
} .

A surface can be queried by providing a query surface. The results of this query will be the final result of the reasoning engine interpreting the surfaces.

@prefix ex: <http://example.org/ns#>.
@prefix log: <http://www.w3.org/2000/10/swap/log#>.

ex:Ghent a ex:City.

# Every city is a human community
(_:S) log:onNegativeSurface {
    _:S a ex:City.
    () log:onNegativeSurface {
        _:S a ex:HumanCommunity.
    }.
}.

# Query
(_:S _:C) log:onQuerySurface {
    _:S a _:C.
}.

@prefix ex: <http://example.org/ns#>.

ex:Ghent a ex:City.
ex:Ghent a ex:HumanCommunity.

More test cases can be found at https://github.com/w3c-cg/rdfsurfaces/blob/master/etc.md.

2. Document Conventions

Within this document, the following namespace prefix bindings to [URI]-s are used:

3. Surface

Surfaces are written as triples where the subject is a list of zero or more blank nodes. The object is a RDF graph or the true or false literal. The blank nodes in the subject list are treated as marks on the object RDF graph. The predicate specifies the kind of surface. Any kind of surface may be used but the following built-in have special semantics:

A surface can contain zero or more other surfaces. These surfaces are then nested. Nested surfaces can share the same [URI]-s and literals (by copying the data), but can’t share blank nodes. Any blank nodes that are written inside a surface (not as subject of an RDF Surface) are to be interpreted as coreferences to the blank node graffiti defined on a parent RDF Surface. If no such parent RDF Surface exists, then it is assumed that the blank node is a coreference to an implicit declared blank node graffiti on the default positive surface.

@prefix ex: <http://example.org/ns#> .

(_:X) ex:myFirstSurface {

    ex:Statement1 a ex:NiceStatement .
    _:X a ex:OtherStatement .
     
    () ex:mySecondSurface {
        _:X ex:is true . 
        _:Y a ex:AnotherStatement .

        (_:X) ex:myThirdSurface {
            _:X ex:is false .
        } .
    }.
} .

3.1. Positive Surface

A positive surface is an RDF graph which claims that an RDF Graph on it is true. This is the current default interperation of RDF Semantics. When no surfaces are provided, an implicit positive surface is assumed in the RDF document.

@prefix : <http://example.org/ns#> .
@prefix log: <http://www.w3.org/2000/10/swap/log#> .

() log:onPositiveSurface {
    :Alice a :Person .
}

@prefix : <http://example.org/ns#> .
@prefix log: <http://www.w3.org/2000/10/swap/log#> .

:Alice a :Person .

The semantics of a positive surface is interpreted as a logical truth:

• An empty positive surface is a tautology

• When a triple is placed on a positive surface, then the surface is valid when the triple is true.

• When two or more triples are placed on a positive surface, then the surface is valid when the logical conjunction of all triples is true.

:Alice a :Person AND 
:Alice :knows :Bob AND (
  :Bob a :Person AND 
  :Bob :knows :Alice
)

@prefix : <http://example.org/ns#> .
@prefix log: <http://www.w3.org/2000/10/swap/log#> .

:Alice a :Person .
:Alice :knows :Bob .

() log:onPositiveSurface {
    :Bob a :Person .
    :Bob :knows :Alice .
} .

When a blank node is marked on a positive surface, then it is interpreted as an existential variable in the scope of the positive surface.

∃ _:X : _:X a :Person AND _:X :knows :Alice

@prefix : <http://example.org/ns#> .
@prefix log: <http://www.w3.org/2000/10/swap/log#> .

(_:X) log:onPositiveSurface {
    _:X a :Person .
    _:X :knows :Alice .
} .

In RDF Surfaces an default positive surface is implicitly assumed for each RDF document. On this default positive surface all existential variables are implicitly quantified. Example 19 can be rewritten as:

@prefix : <http://example.org/ns#> .
@prefix log: <http://www.w3.org/2000/10/swap/log#> .

_:X a :Person .
_:X :knows :Alice .

Note: The use of a positive surface is deprecated and will be removed from future versions of the RDF Surface Primer. A deeply nested positive surface is syntactic sugar for a double nested negative surface.

3.2. Negative Surface

A negative surface is an RDF graph which claims that an RDF Graph on it is false. The intepretation of the negative surface is the negation of RDF Graph on it.

The semantics of a negative surface is interpreted as a logical falsehood:

• An empty negative surface is a contradiction.

• When a triple is placed on a negative surface, then the surface is valid when the triple is false.

• When two or more triples are placed on a positive surface, then the surface is valid when the logical conjunction of all triples is false.

:Alice a :Person AND 
:Alice :knows :Bob AND NOT (
  :Bob a :Person AND 
  :Bob :knows :Alice
)

@prefix : <http://example.org/ns#> .
@prefix log: <http://www.w3.org/2000/10/swap/log#> .

:Alice a :Person .
:Alice :knows :Bob .

() log:onNegativeSurface {
    :Bob a :Person .
    :Bob :knows :Alice .
} .

When a blank node is marked on a negative surface, then it should be interpreted as an existential quantified variable in the scope of the negative surface.

:Alice a :Person AND 
:Alice :knows :Bob AND NOT (
  ∃ _:X :
     _:X a :Person AND 
     _:X :knows :Alice
)

:Alice a :Person AND 
:Alice :knows :Bob AND 
  ( ∀ _:X :
     NOT (
      _:X a :Person AND 
      _:X :knows :Alice
    )
  ) 
)

@prefix : <http://example.org/ns#> .
@prefix log: <http://www.w3.org/2000/10/swap/log#> .

:Alice a :Person .
:Alice :knows :Bob .

(_:X) log:onNegativeSurface {
    _:X a :Person .
    _:X :knows :Alice .
} .

When two or more negative surfaces are nested in a parent negative surface, then it should be interpreted as a logical disjunction.

:Alice a :Person OR
:Bob a :Person OR
:Charly a :Person 

NOT(
  NOT(:Alice a :Person) AND
  NOT(:Bob a :Person) AND
  NOT(:Charly a :Person)
)

@prefix : <http://example.org/ns#> .
@prefix log: <http://www.w3.org/2000/10/swap/log#> .

:Alice a :Person .
:Alice :knows :Bob .

() log:onNegativeSurface {
    () log:onNegativeSurface {
      :Alice a :Person .
    } .

    () log:onNegativeSurface {
      :Bob a :Person .
    } .

    () log:onNegativeSurface {
      :Charly a :Person .
    } .
}.

With combinations of AND and NOT other logical truth function can be build. E.g.

• Disjunction: P ∨ Q : NOT( NOT(P) AND NOT(Q))

• Material implication P → Q : NOT( P AND NOT Q ) .

• Converse implication P ← Q : NOT( NOT(P) AND Q ) .

• Biconditional (P → Q) ∧ (Q → P) : NOT( P AND NOT Q ) AND NOT( Q AND NOT P )

Negative surfaces should also follow the law of double negation elimination: P = NOT(NOT(P)).

3.3. Neutral Surface

A neutral surface makes no claims at all about the graphs written on it. Neutral surfaces can be used, for instance, to store information about RDF Collections. Triples on a neutral surface will not be "executed" in any way by a reasoner. Unless, the triples are copied to a positive or negative surface.

Nesting a neutral surface in another neutral surface will create a nesting of two surfaces.

Writing blank node graffiti on a neutral surface doesn’t have specific semantic meaning.

@prefix : <http://example.org/ns#> .
@prefix log: <http://www.w3.org/2000/10/swap/log#> .
@prefix ldp: <http://www.w3.org/ns/ldp#> .

:Alice a :Person , :BookerPrizeWinner ;
  :name "Alice" .

:Bob a :Person ;
  :name "Bob" .

:Book1 a :Book ;
  :title "Learn RDF in 21 days" ;
  :author :Alice .

:Book2 a :Book ;
  :title "The RDF Surface programming language" ;
  :author :Bob .

() log:onNeutralSurface {
  :BookList1 a ldp:Container ;
    ldp:contains :Book1 , :Book2 .

  :AuthorList1 a ldp:Container ;
    ldp:contains :Alice , :Bob .   
} .

3.4. Query Surface

A query surface is used to return to external applications a single (positive) RDF Surface specified by a graph template. The result is an RDF Surface formed by taking each query solution in the solution sequence, substituting for the variables in the graph template, and combining the triples into a single RDF Surface by set union.

The semantics of a query surface is as follows:

• An empty query surface is an empty template which doesn’t match any triple on the default surface.

• A query surface which contains one or more triples, the triples will create a basic graph pattern that will be matched against all entailments that can be concluded from the default positive surface.

• Blank nodes on a query surface are interpreted as universal quantified variables.

@prefix : <http://example.org/ns#> .
@prefix log: <http://www.w3.org/2000/10/swap/log#> .

(_:S _:P _:O) log:onQuerySurface {
    _:S _:P _:O .
} .

CONSTRUCT {
    ?S ?P ?O .
}
WHERE {
    ?S ?P ?O .
}

@prefix : <http://example.org/ns#> .
@prefix log: <http://www.w3.org/2000/10/swap/log#> .

(_:S _:P _:O) log:onQuerySurface {
    () log:onMyExampleSurface {
        _:S _:P _:O .
    } .
} .

@prefix : <http://example.org/ns#> .
@prefix log: <http://www.w3.org/2000/10/swap/log#> .

CONSTRUCT {
    () log:onMyExampleSurface {
        ?S ?P ?O .
    } .
}
WHERE {
    () log:onMyExampleSurface {
        ?S ?P ?O .
    } .
}

@prefix : <http://example.org/ns#> .
@prefix log: <http://www.w3.org/2000/10/swap/log#> .

(_:G) log:onQuerySurface {
    () log:onMyExampleSurface _:G .
} .

@prefix : <http://example.org/ns#> .
@prefix log: <http://www.w3.org/2000/10/swap/log#> .

(_:P _:O) log:onQuerySurface {
    () log:onNegativeSurface {
        :Alice _:P _:O .
    } .
} .

3.5. Question and Answer Surface

TODO

3.6. Other Surfaces

TODO

4. Examples

TODO

See:

• https://github.com/eyereasoner/eye/tree/master/reasoning/blogic

• https://github.com/eyereasoner/Notation3-By-Example/tree/main/blogic

• https://github.com/eyereasoner/Notation3-By-Example/tree/main/examples